Finite Group Databasebeta

← #13373228

Subgroups — #13373228

Order 1792.

866 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2351C211yesmaximal, characteristic, complemented
2C2×#89798C211yesmaximal, characteristic
3#11731083C211yesmaximal, characteristic
4#11729387C211yesmaximal, characteristic
5#11729387C211yesmaximal, characteristic
6#11729385C211yesmaximal, characteristic, complemented
7C4×#6409C211yesmaximal, characteristic, complemented
8#11729380C211yesmaximal, characteristic, complemented
9#11727371C211yesmaximal, characteristic
10#11727369C211yesmaximal, characteristic
11#11727357C211yesmaximal, characteristic, complemented
12#11727354C211yesmaximal, characteristic, complemented
13#11727310C211yesmaximal, characteristic
14#11727302C211yesmaximal, characteristic, complemented
15#11727232C211yesmaximal, characteristic, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C22×#1143C2211yescharacteristic
20C2×#6455—21no—
21C2×C4×D28C2211yescharacteristic
22C7×#389—21no—
23C7×#389—21no—
24C7×#389C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C7×#382—21no—
27C7×#382C2211yescharacteristic
28C4×C7×C4⋊C4—21no—
29C4×C7×C4⋊C4C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33#89798C418yescomplemented
34#89759—21no—
35C2×#6383C2211yescharacteristic
36C2×#6383C2211yescharacteristic
37C2×#6383C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39#89711C2211yescharacteristic
40#89711C2211yescharacteristic
41#89707C2211yescharacteristic
42C4×C4⋊Dic7C2211yescharacteristic
43C4×C4⋊Dic7C2211yescharacteristic
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45C2×#6377C2211yescharacteristic
46C2×#6377C2211yescharacteristic
47C2×C4×C7⋊C8C2211yescharacteristic
48#89442—21no—
49#89442—21no—
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#22781—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×D28C2311yescharacteristic
63C22×C4×D7—21no—
64C4×C7×Q8—22no—
65C4×C7×Q8—22no—
66C4×C7×Q8—22no—
67C4×C7×Q8—41no—
68C4×C7×Q8—41no—
69C4×C7×Q8C2×C414yes—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4D411yescharacteristic
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C42×C7—21no—
81C2×C42×C7—21no—
82C2×C42×C7—21no—
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×#1143—22no—
87C2×#1143—22no—
88C2×#1143C2×C414yes—
89C2×(C2×D7)⋊C4—21no—
90C2×(C2×D7)⋊C4—21no—
91C2×(C2×D7)⋊C4—21no—
92C2×(C2×D7)⋊C4—21no—
93C2×(C2×D7)⋊C4—21no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7—21no—
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105#6455—44no—
106C4×D28—22no—
107C4×D28C2×C414yes—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95—21no—
111C7×#95—21no—
112C7×#95D411yescharacteristic
113C7×#95D411yescharacteristic
114#6409—22no—
115#6409—22no—
116#6409C2×C414yes—
117C7⋊#95—21no—
118C7⋊#95—21no—
119#6383—41no—
120#6383—41no—
121#6383C2×C414yes—
122#6381C2×C414yes—
123C4⋊(C7⋊C8)—22no—
124#6377—41no—
125#6377C2×C414yes—
126C4×C7⋊C8—41no—
127C2×#440—71no—
128#2354—71no—
129#2351D711yescharacteristic, complemented
130#1936—71no—
131#1934—71no—
132#1921—71no—
133#1918—71no—
134#1880—71no—
135#1880—71no—
136#1836—71no—
137#1804—71no—
138#1803—71no—
139#1771—71no—
140#1763—71no—
141C4×#103—71no—
142C23×D7—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8C22×C412yes—
147C2×C7×Q8#4112yes—
148C2×C7×Q8#4112yes—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2411yescharacteristic
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C2×D28—21no—
164C2×D28—21no—
165C2×D28—21no—
166C2×D28—21no—
167C2×D28—21no—
168C2×D28C22×C412yes—
169C2×C4×D7—24no—
170C2×C4×D7—42no—
171C2×C4×D7—42no—
172C7×C4⋊C4—24no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—41no—
180C7×C4⋊C4—42no—
181C7×C4⋊C4—42no—
182C7×C4⋊C4—42no—
183C7×C4⋊C4—42no—
184C7×C4⋊C4C22×C412yes—
185C42×C7—24no—
186C42×C7—21no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—42no—
191C42×C7—42no—
192C42×C7C22×C412yes—
193C42×C7#4112yes—
194#1143—28no—
195(C2×D7)⋊C4—24no—
196(C2×D7)⋊C4—42no—
197(C2×D7)⋊C4—42no—
198(C2×D7)⋊C4—81no—
199(C2×D7)⋊C4—81no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—42no—
206C4⋊Dic7—81no—
207C4⋊Dic7C22×C412yes—
208C4×Dic7—42no—
209C4×Dic7—42no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8C22×C412yes—
221C2×C7⋊C8C22×C412yes—
222C22×SD16—71no—
223C2×C4×Q8C2×D711yescharacteristic
224C2×C4×Q8C2×D711yescharacteristic
225C2×C4×Q8C2×D711yescharacteristic
226C2×C4×D4—71no—
227C2×#117—141no—
228#440—78no—
229C2×#107—71no—
230C2×C4⋊C8—71no—
231C2×#103—71no—
232C2×#103—71no—
233C2×#103—71no—
234C2×#102—71no—
235C2×#102—71no—
236C2×#102—71no—
237C2×#97—71no—
238C2×#97—71no—
239C2×C4×C8—71no—
240#390—71no—
241#390—71no—
242#389—21no—
243#389—21no—
244#389C2×D711yescharacteristic
245#389C2×D711yescharacteristic
246#388—141no—
247#388—141no—
248#384—71no—
249#382—21no—
250#382C2×D711yescharacteristic
251#380—141no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4—71no—
254C4×C4⋊C4—71no—
255C4×C4⋊C4C2×D711yescharacteristic
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#336—71no—
265#336—71no—
266C23×C7C22×D411yescharacteristic
267C22×D7—22no—
268C22×D7—41no—
269C22×D7—41no—
270C22×D7—41no—
271C22×D7—41no—
272C22×D7—41no—
273C22×D7—41no—
274C7×Q8—22no—
275C7×Q8—22no—
276C7×Q8—22no—
277C7×Q8#11714yescomplemented
278C2×C4×C7—22no—
279C2×C4×C7—21no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—22no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7C4×D412yes—
306C2×C4×C7C4×D412yes—
307C2×C4×C7AGL(1,ℤ/8)12yes—
308C2×C4×C7C23×C411yescharacteristic
309C2×C4×C7C2×#4111yescharacteristic
310C2×C4×C7C2×#4111yescharacteristic
311C2×C4×C7C2×#4111yescharacteristic
312C2×C4×C7H(5,𝔽2)11yescharacteristic
313C2×C4×C72−1+411yescharacteristic
314C2×Dic7—22no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323D28—22no—
324D28—22no—
325D28—41no—
326D28—41no—
327C4×D7—48no—
328C7⋊C8—22no—
329C7⋊C8—22no—
330C7⋊C8—41no—
331C7⋊C8—41no—
332C22×Q8C22×D711yescharacteristic
333C22×D4—71no—
334C23×C4—141no—
335C2×SD16—74no—
336C2×SD16—142no—
337C2×SD16—142no—
338C22×C8—71no—
339C22×C8—71no—
340C22×C8—71no—
341C22×C8—71no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—41no—
346C4×Q8—41no—
347C4×Q8C4×D714yes—
348C4×D4—74no—
349C4×D4—142no—
350#117—284no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—71no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4C22⋊D711yescharacteristic
362C2×C4⋊C4C22⋊D711yescharacteristic
363C2×C4⋊C4C22×D711yescharacteristic
364C2×C4⋊C4C22×D711yescharacteristic
365C2×C4⋊C4C22×D711yescharacteristic
366C2×C22⋊C4—141no—
367C2×C22⋊C4—141no—
368C2×C22⋊C4—141no—
369C2×C22⋊C4—141no—
370C2×C22⋊C4—141no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—141no—
375C2×C42—141no—
376C2×C42C22×D711yescharacteristic
377C2×C42C22×D711yescharacteristic
378C2×C42C22×D711yescharacteristic
379#107—74no—
380C4⋊C8—142no—
381#103—74no—
382#103—142no—
383#103—142no—
384#102—74no—
385#102—281no—
386#102—281no—
387#97—74no—
388#97—281no—
389C4×C8—281no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—21no—
394#95—141no—
395#95—141no—
396#95C22⋊D711yescharacteristic
397#95C22⋊D711yescharacteristic
398C22×C7C2×C4×D411yescharacteristic
399C22×C7#54611yescharacteristic
400C22×C7#54811yescharacteristic
401C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
402C22×C7#57511yescharacteristic
403C22×C7#57711yescharacteristic
404C22×C7#57811yescharacteristic
405C2×D7—42no—
406C2×D7—42no—
407C2×D7—42no—
408C2×D7—81no—
409C2×D7—81no—
410C2×D7—81no—
411C2×D7—81no—
412C4×C7—24no—
413C4×C7—22no—
414C4×C7—22no—
415C4×C7—22no—
416C4×C7—42no—
417C4×C7—41no—
418C4×C7—41no—
419C4×C7—41no—
420C4×C7—81no—
421C4×C7—81no—
422C4×C7C2×#11711yescharacteristic
423C4×C7#52011yescharacteristic
424C4×C7#55211yescharacteristic
425C4×C7#55311yescharacteristic
426Dic7—42no—
427Dic7—81no—
428Dic7—81no—
429C24—141no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8C2×C4×D712yes—
434C2×Q8#115912yes—
435C2×Q8#116112yes—
436C2×D4—72no—
437C2×D4—141no—
438C2×D4—141no—
439C2×D4—141no—
440C2×D4—141no—
441C2×D4—141no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—141no—
447C22×C4—144no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—282no—
451C22×C4—282no—
452C22×C4C2×C22⋊D711yescharacteristic
453C22×C4C2×C22⋊D711yescharacteristic
454C22×C4C2×C22⋊D711yescharacteristic
455C22×C4C2×C22⋊D711yescharacteristic
456C22×C4C2×C22⋊D711yescharacteristic
457C22×C4C2×C22⋊D711yescharacteristic
458C22×C4C23×D711yescharacteristic
459SD16—148no—
460C2×C8—72no—
461C2×C8—72no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8—141no—
471C2×C8—141no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—41no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—72no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4—282no—
491C4⋊C4—561no—
492C4⋊C4C2×C4×D712yes—
493C22⋊C4—144no—
494C22⋊C4—282no—
495C22⋊C4—282no—
496C22⋊C4—561no—
497C22⋊C4—561no—
498C42—24no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—42no—
504C42—42no—
505C42—282no—
506C42—282no—
507C42C2×C4×D712yes—
508C42#115712yes—
509C2×C7#235211yescharacteristic
510C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
511C2×C7#294711yescharacteristic
512C2×C7#330811yescharacteristic
513C2×C7#332111yescharacteristic
514C2×C7#336811yescharacteristic
515C2×C7#337711yescharacteristic
516D7—82no—
517C23—142no—
518C23—281no—
519C23—281no—
520C23—281no—
521C23—281no—
522C23—281no—
523C23—281no—
524C23C22×C22⋊D711yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—22no—
528Q8#645514yescomplemented
529D4—142no—
530D4—142no—
531D4—281no—
532D4—281no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—142no—
561C2×C4—288no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4C4×C22⋊D712yes—
571C2×C4C4×C22⋊D712yes—
572C2×C4#651212yes—
573C2×C4C22×C4×D711yescharacteristic
574C2×C4C2×#115711yescharacteristic
575C2×C4C2×#115911yescharacteristic
576C2×C4#654811yescharacteristic
577C2×C4C2×#116111yescharacteristic
578C2×C4#655111yescharacteristic
579C8—142no—
580C8—142no—
581C8—281no—
582C8—281no—
583C7#2278111yescharacteristic, complemented, Sylow
584C22—282no—
585C22—282no—
586C22—282no—
587C22—561no—
588C22—561no—
589C22—561no—
590C22—561no—
591C22C2×C4×C22⋊D711yescharacteristic
592C22#9049711yescharacteristic
593C22#9050811yescharacteristic
594C22C2×#651211yescharacteristic
595C22#9051311yescharacteristic
596C22#9052811yescharacteristic
597C22#9052911yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—24no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—42no—
606C4—81no—
607C4—81no—
608C4—282no—
609C4—561no—
610C4—561no—
611C4C2×#645511yescharacteristic
612C4#9019511yescharacteristic
613C4#9020411yescharacteristic
614C4#9020511yescharacteristic
615C2—562no—
616C2#1173960611yescharacteristic
617C2C4×#651211yescharacteristic
618C2#1173999211yescharacteristic
619C2#1174098111yescharacteristic
620C2#1174098411yescharacteristic
621C2#1174113511yescharacteristic
622C2#1174114511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2351C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#2351D7
hypocenterC7#22781
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C7×#2351C2semidirect
6. #11729385C2semidirect
7. C4×#6409C2semidirect
8. #11729380C2semidirect
11. #11727357C2semidirect
12. #11727354C2semidirect
14. #11727302C2semidirect
15. #11727232C2semidirect
33. #89798C4semidirect
129. #2351D7semidirect
277. C7×Q8#117semidirect
528. Q8#6455semidirect
583. C7#22781semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C7×#23512
C2×#897982
#117310832
#117293872
#117293852
C4×#64092
#117293802
#117273712
#117273692
#117273572
#117273542
#117273102
#117273022
#117272322
#227817